# Divisibility Rules A. Divisibility by 2, 4, 8, 16, 32.

A number is divisible by 2, 4, 8, 16, 32,.. 2 when the number formed by the last one, two, three, four, five…n digits is divisible by 2, 4, 8, 16, 32,..2 respectively.

Example: 1246384 is divisible by 8 because the number formed by the last three digits i.e. 384 is divisible by 8. The number 89764 is divisible by 4 because the number formed by the last two digits, 64 is divisible by 4.

B. Divisibility by 3 and 9

A number is divisible by 3 or 9 when the sum of the digits of the number is divisible by 3 or 9 respectively.

# LOGARITHM

If a is a positive real number other than 1, and am = x, then we can write: m = logax, and we say that the value of log x to the base a is m.
Examples:

(i) 104 = 10000     log10 10000 = 4.

(ii) 35 = 243     log3 243 = 5.

(iii) (0.1)3 = 0.001     log (0.1) (0.001) = 3.

Properties of Logarithms

# Remainder Theorems

It is uncanny how children pick up a lot of small habits and beliefs of their parents. Even the ones that rebel against their parents bear the subconscious resemblance to their father or mother. There is a lesson for instructors in this. It is important for them to realize that students mirror their feelings about CAT. If the instructor expresses or feels that CAT is tough, or is fearful about CAT, his students will mirror the same feeling and will be less confident. If the instructor is brazen and casual about the paper and scoffs the competition, his students would reflect the same feelings. Also, it is so necessary to have unflinching faith in one’s students. I still remember that during my school days my mother used to proudly proclaim that I was an intelligent kid. I was barely scrapping passing marks in the school exams. If truth be told I was at the bottom of the class, but my mother had her blinkers on. And because of my mother, I also believed that I was second to none. It was only years later, during my boards exams, that I took to studying seriously, and managed to outperform everyone else. I don’t know if it was my mother’s blind love for me or that she could see some bright spark in me that made her claim my intelligence but it really had great effect on my attitude. And attitude, in an exam like CAT, is everything.

Well, it’s almost everything. A keen attitude towards CAT subjects’ fundamentals also plays an important role in one’s performance.

# Counting in Geometry

(a) In a plane, if there are n points of which no three are collinear, then

1. The number of straight lines that can be formed by joining them is nC2.
2. The number of triangles that can be formed by joining them is nC3.
3. The number of polygons with k sides that can be formed by joining them is nCk.

(b) In a plane, if there are n points out of which m points are collinear, then

1. The number of straight lines that can be formed by joining them is nC2mC2 + 1.
2. The number of triangles that can be formed by joining them is nC3mC3.
3. The number of polygons with k sides that can be formed by joining them is nCkmCk.

(c) The number of diagonals of an n sided polygon is nC2 – n = n × (n – 3)/2

I am dividing this method into four parts and we will discuss each part one by one:

a. Last two digits of numbers which end in one
b. Last two digits of numbers which end in 3, 7 and 9
c. Last two digits of numbers which end in 2
d. Last two digits of numbers which end in 4, 6 and 8

Before we start, let me mention binomial theorem in brief as we will need it for our calculations.

Last two digits of numbers ending in 1

What are the last two digits of 31786?

There have been questions, in recent years CAT’s quant and data interpretation sections, based on some mathematical strategy games. These questions do not require much mathematical expertise but seek for your hidden intelligence. Just like many things which you might have studied at your degree college are practical i.e. useful but not practicable i.e. feasible. Also in a business school, you learn so much business jargon and tactics but how much is practicable you learn only after getting out of school and enter the business. The exactly same way you may know ‘n’ amount of mathematics but that sharp intelligence, aptitude to solve problems is the key which CAT is looking for.

All of us know that mathematics can be learned but there are no courses for learning intelligence (* leave Artificial Intelligence for the time being). But we can nurture and nourish intelligence by using it. So let’s do it.

Hello World!! Now that you are well set on your journey of grammar, here’s a small lesson for you on one of the most frequent error zones in grammar- Modifiers.

Oh, by the way! Do you like coffee? It might not be a bad idea to keep a steaming mugful by your side as you read the lesson. Sip along!

Modifiers- Dangling and Misplaced:
A modifier describes, clarifies, or gives more detail about a word or a word group. A dangling modifier is a phrase that modifies the wrong word or phrase because of the absence of the word or phrase it is supposed to modify. In other words, the modifier is left “dangling”. A misplaced modifier is placed incorrectly in the sentence such that it modifies the wrong word and makes the sentence sound illogical.

The rules of business can be applied to any competitive environment. Just as businesses keep trying to achieve an extra edge and sell more and more products so do students preparing for competitive exams such as CAT 2018. The focus should always be on trying to achieve more and more. For CAT aspirants, this is the golden rule- always be focused on your goal. That means giving away all your distractions: that movie which other friends are going to watch, those phone calls, whatsapp, facebook, or twitter, those sms, those days of boredom when you do not want to study, and so much more. For every second, minute, hour or day that you lose, someone is trying to get ahead of you. The fact is that you are not going to become a CAT cracker overnight; you will have to grow inch by inch. And the other fact is that it is easier to increase your percentile to 95 or 97 from 70 or 80 percentile but it is much much tougher to increase it further to 99+. So the more you are at the top, the harder it is to grow further. And it takes focus and dedication climb day by day.

I am very fond of today’s chapter. These tricks are used by me off an on and I teach them in my first class also. The result is that all my students are able to work out calculations mentally by the end of the class. The sad fact is that they do not continue it once they get home, which is a pity. If they keep up the practise for 3 or 4 months, they would see themselves doing wonders, as I saw with myself. But practicing it day in and day out again takes focus and vision of one’s goal.

• Nice instruction, thanks

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